\begin{align*} \mu & = E[X] = \sum_{x=1}^{\infty} {x(1-p)^{x-1}p}\\ & = p \sum_{x=1}^{\infty} {x(1-p)^{x-1}}\\ & = p \frac{1}{(1-(1-p))^2}\\ & = p \frac{1}{p^2} = \frac{1}{p} \end{align*}

\begin{align*} \sigma^2 & = E[X(X-1)] + \mu - \mu^2 \\ & = \sum_{x=1}^{\infty} {x(x-1)(1-p)^{x-1}p} + \mu - \mu^2 \\ & = (1-p)p \sum_{x=2}^{\infty} {x(x-1)(1-p)^{x-2}} + \frac{1}{p} - \frac{1}{p^2}\\ & = (1-p)p \frac{2}{(1-(1-p))^3} + \frac{1}{p} - \frac{1}{p^2}\\ & = \frac{1-p}{p^2} \end{align*}